1. Introduction

Delays are inherent in many physical and technological systems. In particular, pure delays are often used to ideally represent the effects of transmission, transportation, and inertia phenomena. Delay differential equations constitute basic mathematical models of real phenomena, for instance in biology, mechanics, and economics (cf., e.g., [1–17] and references therein). Stability analysis of delay differential equations is particularly relevant in control theory, where one cause of delay is the finite speed of communication. There have been a lot of results on the study of stability of delay differential equations. For example, we can see many earlier results on this issue from Burton's book [2]. Recently, in 2004, Butcher et al. [4] studied the stability properties of delay differential equations with time-periodic parameters. By employing a shifted Chebyshev polynomial approximation in each time interval with length equal to the delay and parametric excitation period, the system is reduced to a set of linear difference equations for the Chebyshev expansion coefficients of the state vector in the previous and current intervals. In 2005, Wahi and Chatterjee [16] used Galerkin-projection to reduce the infinite dimensional dynamics of a delay differential equation to one occurring on a finite number of modes. In 2009, Kalmár-Nagy [7] demonstrated that the method of steps for linear delay differential equation together with the inverse Laplace transform can be used to find a converging sequence of polynomial approximants to the transcendental function determining stability of the delay equation. Most recently, Berezansky and Braverman [3] gave some explicit conditions of asymptotic and exponential stability for the scalar nonautonomous linear delay differential equation with several delays and an arbitrary number of positive and negative coefficients.

This paper is concerned with the following differential equations with nonconstant delay: where , , , and and with ( is a positive constant) are continuous functions. We aim at giving general criterion for the zero solution of this delay equation being uniformly stable and asymptotically stable.

2. Main Result

Denote by the Banach space of continuous functions from to with the sup-norm

We consider (1) for with the initial conditions (for any ) where .

For an initial function , we denote by the solution of (1) such that (4) holds.

Definition 1. The zero solution of (1) is said to be stable if for any and , there exists such that if then The zero solution of (1) is uniformly stable if the above is independent of .

Definition 2. The zero solution of (1) is said to be asymptotically stable if it is stable and if for any , there exists such that if then

Theorem 3. Assume that (1)the zero solution to (1) is unique;(2)if is nontrivial function and is nontrivial in any interval , then for a constant ; (3); (4)if , then where .
Then the zero solution of (1) is uniformly stable.

Proof. For each , we set and when is a nontrivial function and is nontrivial in any interval (), we set From (3) and (2), it follows that for every , there exists such that and when is a nontrivial function and is nontrivial in any interval (), such that We claim that for any and , if then which means that the zero solution of (1) is eventually uniformly stable. Actually, if this is not true, then there exist and a solution to (1) with and such that there is a , Define Then, together with (21) and (22), we obtain and, for , and for arbitrary , there exists such that Therefore, This implies that In fact, if then by (23)–(25), we have It is not hard to see that we can choose and above to make have constant sign in .
Case I. When or where is a positive real number.
In this case, if , then which contradicts with (28). Moreover, if for a positive real number , then it is clear that we can require . Hence, which contradicts with (28) too.
Consequently, in this case we have the following observation.
Case I-1. If , then we deduce by (23), (24), (1), and (11) that This is clearly impossible.
Case I-2. If , then we deduce by (23), (24), (1), (11), and (14) that This is clearly impossible too.
Therefore, in this case, the zero solution of (1) is eventually uniformly stable. This, together with assumption (1), implies that the zero solution of (1) is uniformly stable.
Case II. is a nontrivial function and is nontrivial in any interval ().
In this case, by virtue of (1), and assumption (2), (12), (13), and (16), we get which contradicts with (28).
Consequently, in this case we have the following observation,
Case II-1. If , then we deduce by (23), (24), (1), (11), (12), (14), and (15) that
This is a contradiction.
Case II-2. If , then we deduce by (23), (24), (1), (11), (12), (14), and (15) that This is a contradiction too.
Therefore, in this case, the zero solution of (1) is eventually uniformly stable. This, together with assumption (1), implies that the zero solution of (1) is uniformly stable.

Theorem 4. Assume that (1)the zero solution to (1) is unique;(2)if or for a positive real number , then (3)if is nontrivial function and is nontrivial in any interval   , then for a constant ;(4); (5)if , then where . Then the zero solution of (1) is asymptotically stable.

Proof. It follows from Theorem 3 that the zero solution of (1) is uniformly stable; that is, for arbitrarily given and , there exists such that if then Next, we will prove that First, we show that Suppose that this is not true. Then Hence, for the arbitrarily given there exist and such that or Let us now consider
Case I. When or for a positive real number , we obtain by assumption (2), (46), (50), and (53) This implies that which contradicts with (53).
Case II. When is a nontrivial function and is nontrivial in any interval (), we obtain by assumptions (3), (46), (50), and (53) This, together with assumption (2), implies that which contradicts with (53).
Moreover, in a similar way, we can prove that is impossible.
Therefore, (48) is true.
Based on (48), we will show that Actually, if this is not true, that is, then by (48) we see that there are with and two sequences and such that and for , By the same reason as that in the proof of Theorem 3, we know that Define , , , and as those in the proof of Theorem 3. Then when is large enough, we have
Case I. When or where is a positive real number.
Case I-1. If , then we deduce that This is impossible.
Case I-2. If , then we obtain This is clearly impossible too.
Consequently, (60) is true in this case.
Case II. When is nontrivial function and is nontrivial in any interval ().
Case II-1. If , then we deduce that This is a contradiction.
Case II-2. If , then we obtain This is a contradiction too.
Therefore, (60) is true in this case. So, (60) holds truly. This means that the zero solution of (4) is asymptotically stable.